Here we should note that theorem REF  does not depend on a particular excitation function \(\mu (t)\)  and this detail motivates it's use for fitting more general Hawkes processes than those with exponential decay. In practice the log-likelihood is often preferred which was introduced in [1]} as follows.
\(\log (L)=\sum _{i=1}^k \log \left(\lambda (t_i)\right) - \int _0^T \lambda (u) du\) 
